A Dunkl-Williams Inequality and the Generalized Operator Version

Abstract

C.F. Dunkl and K.S. Williams (Am. Math. Mon. 71, 53–54 (1964)) showed that for any nonzero elements x, y in a normed linear space $\mathcal{X}$

$$\biggl \Vert \frac{x}{ \Vert {x}\Vert }-\frac{y}{ \Vert {y}\Vert }\biggr \Vert \leq\frac{4 \Vert {x-y}\Vert }{ \Vert {x}\Vert + \Vert {y}\Vert }. $$

Recently, J. Pečarić and R. Rajić (J. Math. Inequal. 4, 1–10 (2010)) gave a refinement and, moreover, a generalization to operators $A,B \in\mathcal{B}(\mathcal{H})$ such that |A|, |B| are invertible as follows:

$$\bigl|A|A|^{-1}-B|B|^{-1}\bigr|^2 \le|A|^{-1}\bigl(p|A-B|^2+q\bigl(|A|-|B|\bigr)^2\bigr)|A|^{-1} $$

where p,q>1 with $\frac{1}{p}+\frac{1}{q}=1$.

In this note, we review some results concerning the Dunkl-Williams inequality and the generalization of the operator version of J. Pečarić and R. Rajić.